A Physically-Motivated Triode Model for Circuit Simulations

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Proc. of the 14th Int. Conference on Digital Audio Effects (DAFx-11), Paris, France, September 19-23, 2011 Proc. of the 14th International Conference on Digital Audio Effects (DAFx-11), Paris, France, September 19-23, 2011

A PHYSICALLY-MOTIVATED TRIODE MODEL FOR CIRCUIT SIMULATIONS

Kristjan Dempwolf and Udo Zölzer Dept. of Signal Processing and Communications, Helmut Schmidt University Hamburg Hamburg, Germany kristjan.dempwolf|[email protected]

ABSTRACT bution to our triode model, we start with a view to simple vacuum diodes. A new model for triodes of type 12AX7 is presented, featuring simple and continuously differentiable equations. The description is physically-motivated and enables a good replication of the 2.1. Diode Characteristics grid current. Free parameters in the equations are fitted to refer- The most simple vacuum tube, the so-called vacuum diode, con- ence data originated from measurements of practical triodes. It is sists of only two electrodes which are mounted in a vacuum cylin- shown, that the equations are able to characterize the properties der. One electrode, the cathode (K), is heated and hence electrons of real tubes in good accordance. Results of the model itself and are emitted. The second electrode is called the anode or plate (A) when embedded in an amplifier simulation are presented and align and is designed to collect the emitted electrons. Since electrons well. can not be emitted by the cold anode the assumption of unilateral conductivity is valid. In unheated condition, no current flow is 1. INTRODUCTION possible.

The theory of vacuum tubes was already discussed at full length 2.1.1. Initial Velocity Current many decades ago and the best books on these devices were still written in the 1930s to 1950s [1, 2, 3, 4]. The idea of model- Without an external voltage applied to the anode, and even for a ing tubes intends not to challenge the correctness of the traditional slightly negative anode with respect to the cathode, some of the formulations that can be found in those books. It is more the task faster electrons reach the anode and a small current flow is ob- of emulating the behavior of tubes at their boundaries of purpose. served. A small negative voltage must be applied in order to sup- While the first tube models were inspired by the desire for SPICE- press the current flow. The current through the diode I follows the based simulations of hi-fi circuits, the newer approaches are rather exponential equation motivated by real-time simulations and guitar amplifier circuits. V E These amplifier designs gain their special sound by intentionally I = I0 e T , (1) · provoking a distortion of the instruments’ signal and thus have just a little in common with ordinary amplifier theory. The recent ac- with anode voltage V , thermal voltage ET (in Volt) and current I0 tivities towards tube modeling consider the operation in overdriven at zero voltage. amplifiers [5]. With the digital simulation of analog audio circuits in mind, 2.1.2. Space-Charge Current the investigation of tube models is a natural next step. Any im- provement in the critical parts promises better performance of the When a positive voltage is applied from anode to cathode, the complete simulation. Such improvements include challenges such emitted electrons are pulled by the anode and an increasing cur- as accuracy, computational complexity and robustness. rent flow is observed. But only a part of the emitted electrons The paper is organized as follows: The second section will reach the anode. Most of them have only low velocity, they stay review the physical fundamentals of vacuum tubes. This is neces- near the hot cathode and form the space charge, a cloud of nega- sary, because for a successful modeling of tubes or tube circuits, a tive charges. The resulting current depends on the anode voltage good knowledge of the theory is required. The tube experts may and is self-limited by reason of the electric field of the (negative) skip this part. In Section 3 practical tubes are discussed. After charges. This is expressed by the Langmuir-Child equation this some modeling approaches are reviewed in Section 4 and the 3 I = G V 2 , V > 0, (2) new contribution is introduced in Section 5. Finally, in Section · 6, some results are presented, utilizing the model equations in the with perveance G, a constant that only depends on the geometrical state-space model of a common amplifier stage. construction of the tube.

2. VACUUM TUBE BASICS 2.1.3. Saturation Current In this section we give a very brief introduction to the fundamen- Beyond a certain voltage, all emitted electrons are drawn to the tals of vacuum tubes. For detailed exploration we refer to the his- anode. Saturation appears, that means, the current remains nearly toric standard literature [1, 2, 3, 4]. Because of a certain contri- constant when the external voltage is increased further.

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2.2. Triode Characteristics 8 3 → Triodes have a third electrode placed between anode and cathode, → which is constructed as a wire mesh and therefor called grid (G). 6 A voltage applied to the grid is able to control the flow of elec- 2 trons from the cathode: a negative grid voltage causes an electric field that counteracts with the electric field from the anode, the 4 cathode current is reduced. A higher negative grid voltage inhibits the current flow thoroughly. Small changes in grid voltage cause 1

high changes in current flow, thus it is possible to use triodes for anode current2 in mA anode current in mA amplification. For clarity we define the following conventions: Va is the 0 anode- and Vg the grid voltage (both referred to the cathode potential), Ik, Ia and Ig are the cathode, anode and grid currents, 0 100 200 300 0 100 200 300 respectively. We define Ig + Ia = Ik where Ia and Ig are directed anode voltage in V anode voltage in V into the device. → → For the introduction of a grid between anode and cathode it (a) exponent variation (b) Veff < 0 range is a common prospect to replace the three-terminal triode by a two-terminal one [2], where the electrode is placed at the grids’ Figure 1: Anode current calculated with Langmuir-Childs formula position and loaded with the effective voltage, (dotted) and measurements from a RSD 12AX7 triode (solid) in a qualitative plot. 1 V = Vg + Va . (3) eff µ and anode current were measured at the same time in good res- The amplification factor µ states about how much higher the anode olution for a fine mesh of discrete Vg and Va voltages. Since current is influenced by the grid than by the anode voltage. The tube circuits are normally operated with AC, the measurements current Ik is calculated analog to equation (2), were not performed under static conditions, but using switched 3 voltage supplies and triggered meters. The two DC supplies for Ik = G (Veff) 2 ,Veff > 0. (4) · Va and Vg were switched on at the same time and the currents Equation (4) is valid for the “normal” operating condition, were measured for a short time period. From the stored current where the grid is slightly more negative than the cathode, while the values the median was chosen to be the final value. The dis- tance between the discrete measurement points was chosen small anode potential is very high (e.g. Vg = 2 V and Va = 300 V). As for the diode we speak of the space-charge− region. The grid enough, that a subsequent interpolation is not required. In con- sideration of the constrains given by maximum power consump- draws no current (Ig = 0) and consequentially anode and cathode tion and maximum anode current, the observed working range was current are equal, Ia = Ik. Va = 20 V to 300 V and Vg = 5 V to 3 V. The discrete steps for As a special case we consider the operation with a positive − the grid were ∆Vg = 0.1 V, Vg < 1 V and less dense for higher grid. Since the grid now likewise attracts electrons, a positive cur- | | values, and ∆Va = 20 V for the anode voltage. This manually per- rent Ig arises and hence the cathode current divides into anode and grid current. Most books do not respect this case in detail, because formed procedure is highly time-consuming (ca. 8h net./system), the flow of grid current introduces distortions and thus do not sat- so only three triode systems were tested as a start. isfy the classical amplifier theory anymore. The positive grid is At first we settle for evaluating the measurements only quali- irrelevant for the design of linear hi-fi tube amplifiers. Fact is, that tatively, some complete datasets will be shown in Section 5.5. most guitar amplifier designs since the 1960s are operated under these conditions and that the grid current is responsible for some 3.2. Observations on the Anode Current distortion effects like the blocking distortion [6, 7]. For a correct simulation the inclusion of the grid current can play an important Figure 1 opposes qualitatively a measurement from an old 12AX7 role, especially if coupled stages in a cascade are examined. tube (RSD) and Langmuir-Child’s law. While the measurement in principle follows the formula, two differences are visible: First, the theoretic 3 -exponent does not fit perfectly for real triodes, see 3. PRACTICAL TUBES 2 1(a). In the plotted example, the slope of the measured curve is < 3 Real tubes depart occasionally clearly from the idealized formula- lower, representing an exponent 2 . tions given in Section 2 as well as from the information found in The second difference can be found at the bottom of the curve. the manufacturer’s data sheets. Aging effects, small constructive Equation (4) yields Ik = 0 for Veff = 0, and is not defined for variations and origin have influence on the characteristics, to name voltages Veff < 0. The measurement offers a small current for this a few reasons. Even tubes from the same type and same manu- case and a smooth transition towards zero current. This is depicted facturer may show up to 20 % deviation of each other. Exemplary in Figure 1(b), where the highlighted region tags the range Veff < measurement results are discussed e.g. in [8, 9]. 0. We will resume these points later on in Section 5.

3.1. Measurements 3.3. Observations on the Grid Current To have reliable reference data, numerous measurements on stan- The measured grid current increases when moving from small neg- dard triodes of type 12AX7 were performed by the authors. Grid ative voltages towards the ordinate following an exponential shape,

DAFX-2 DAFx-258 Proc. of the 14th Int. Conference on Digital Audio Effects (DAFx-11), Paris, France, September 19-23, 2011 Proc. of the 14th International Conference on Digital Audio Effects (DAFx-11), Paris, France, September 19-23, 2011 see Figure 2(a). For positive voltages the current increases less 5. NEW TRIODE MODEL steep, but may reach significant values (Figure 2(b)). The grid current Ig is highly influenced by Vg, but has only small dependency Based on the observations from Section 3 a new triode desription on Va. is deployed. We follow the idea, that the model in general has to be physically motivated, i.e. the formulations have to follow the traditional equations as explained in the second section. Furthermore 20 1 the model has to be adaptable, so that simulations can be fitted individually to a selected tube. Last but not least, formulations with 0.8 low complexity are desirable, to enable real-time applications. → 10 → A 0.6 µ 5.1. Cathode Current in in mA g g 0.4 I 0 I The exponent in equation (4) may differ from the theoretical value 3 0.2 2 , as already stated in Section 3.2. This can be explained by the fact, that the exponent is calculated for simplified and ideally con- 0 structed triodes and thus may differ from practical ones. -4 -2 0 0 0.5 1 Kniekamp focused in a historic study on the exponent and specified several counteracting reasons for the deviation [13], say- grid voltage in V grid voltage in V → → ing that the exponent may be both smaller or greater than 1.5. Re- (a) grid negative (b) grid positive ich [1] identified the exponent generously to be approximately in the range 1.2 to 2.5. Figure 2: Grid current measured for 12AX7 triode (same tube). In fact, this was already part of other triode models, Koren for example assumed a fixed exponent of 1.4 in his model [10]. In the upcoming model the exponent will be parametrized with γ. This leads to the first approximation for the cathode current

γ 4. MODELING OF TUBES Ik G (V ) ,V > 0. (5) ≈ · eff eff

In the last 20 years about a dozen different tube models for SPICE Note that the transition for Veff 0 is not considered in this equa- simulation tools were invented. These mathematical expressions tion. ≤ are also widely used in digital audio effect algorithms. For application of those SPICE models we refer to a recent book [9] and the original papers. 5.2. Grid Current Basically we can distinguish between physical models, which With a view to the mechanical construction it is obvious that the are based on the physical tube equations and heuristic or phe- relation between grid and cathode can be modeled roughly as a nomenological models, which have no physical foundation. A dif- vacuum diode. The variables may differ considerably since the ferent distinction can be made by classifying models by purpose of grid electrodes’ construction is distinct from the solid anode. But use. While the first tube models were motivated by SPICE-based in general any N-electrode arrangement will show an initial veloc- hi-fi circuit analysis, the newer models are explicitly invented for ity current and space-charge effects, cf. [2]. guitar amplifier simulation. In the following we will review three From the measurements we found, that the grid current Ig is approaches briefly to get an introduction to the subject. highly influenced by the grid voltage, but has only small depen- One of the most popular models is a phenomenological de- dency on the anode voltage. This leads to a simplified approxima- scription invented by Koren [10]. In his approach the triode is tion for Ig = f(Vg), composed from basic SPICE components, namely current sources, ξ Ig Gg Vg (6) resistors and diodes and a mathematical description. The work ≈ · features a library with popular tubes. Koren’s model gives good with the grid perveance Gg and exponent ξ. Similar observations results for simulations with negative grid. are specified in [12] and supported by the literature [2, 4]. Note A recent model was proposed by Cardarilli et al. [11]. In their that equation (6) corresponds to the space-charge law, equation (4), but with possibly deviant exponent. Near to Vg 0 the grid formulation the “tube constants” (e.g. µ, G) are exchanged by ≈ − 3rd-order polynomials, then the model (with its many unknowns) current was measured to resemble an exponential curve. This can is fitted to measurement data. The presented results, including the be explained by the initial velocity current, as in equation (1). simulation of a guitar amplifier, are very promising. Nevertheless the formula has lack of physical interpretation, because the poly- 5.3. Smooth Transition nomial for the perveance, which is the only real constant, has a high excursion. To create a smooth transition between piecewise functions various Cohen and Helie [12] extended Koren’s model by a more real- approaches are possible. For our model we decided to use a com- istic grid current, using a piecewise-defined function with a linear bination of exponential function and logarithm. As introduction part, a second-order polynomial and a smooth transition. They we examine a simple function f(x) with performed current measurements in a similar manner, followed by x , x > 0 a bilinear interpolation. Based on these data a fitting is executed, f(x) = (7) identifying Koren’s parameters individually for three triodes. (0 , x < 0.

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f(x) g(x) h(x) 3

2 15 1

→ 10 -2 0 2 4

in mA 5 x k → I Smoothing function. Figure 3: 0 5 Obviously there is a knee at x = 0. To smoothen this discontinuity 300 we consider a second function 0 200 100 −5 x U in V → 0 U in V → g(x) = log(1 + e ) (8) g a showing the same tendencies as equation (7) for greater values of x . With increasing x the result tends to g(x) log(ex) = x, Figure 4: Fitting of equation (10) to measured Ik. |while| the exponential and thus the function value≈ approaches zero in x direction. −The curve shape can furthermore be adapted towards the linear function using an additional factor C, leading to

C x 1 3 h(x) = log(1 + e · ) . (9) · C 2

This relation is illustrated in Figure 3, where equations (7), (8) → and (9) are plotted (adaption factor was chosen C = 2). 1 With this mathematical trick the smoothing of our modeling in mA g equations comes to mind. The idea is not new, by taking a closer I 0 look to Koren’s model, we figure out equation (8) in the expression for the anode current. Evidently equation (9) with the extension is −1 5 more flexible. 300 0 200 5.4. Final Equations 100 −5 U in V → 0 U in V → The derivations for the cathode and the grid current are now as- g a sembled and the smoothing is applied to our formulations for Ik and Ig. Enhancing equation (5) and equation (6) yield the final Figure 5: Fitting equation (11) to measured Ig. formulations:

1 γ Using a curve fitting algorithm like sftool it is now possible 1 1 to adapt the parameters to the measurement data. Figure 4 shows Ik = G log 1 + exp C µ Va + Vg (10) · · · · C ! both measurement data and fitting curve of the cathode current for a 12AX7 triode, displayed as a 3-D plot. The surface represents the ξ 1 analytically computed characteristics according to equation (10) Ig = Gg log 1 + exp Cg Vg + Ig0, (11) · · · Cg while the measurements are given as black dots. The influence of the grid voltage on the current is obvious. Figure 5 displays the with the perveances G and Gg, exponents γ and ξ and the adaption same for the grid current, revealing that Ig is almost independent factors C and Cg. For the grid current a constant Ig0 is added due of Va, as mentioned before. It can be asserted that the measure- to stability reasons. The anode current subsequently has to be the ment dots align well with the surfaces for both plots. difference of both, giving The fitting results for three systems are given in Table 1. For- tunately, G, µ and γ reflect the standard values that can be found Ia = Ik Ig. (12) − in the books in a satisfying approximation. These equations still reveal the well-known physics but with some The 3-D plots are nice for getting an idea of the dependencies, degree of freedom. Furthermore, they are continuously differen- although parametric curves allow for a better inspection of the re- tiable and have no discontinuities. sults. In Figure 6 the complete characteristics of a measured triode are compared to the results from the fitting. The influence of grid and anode voltage on the currents is clearly visible and accurately 5.5. Fitting to the Measurements replicated by the model. The model equations feature four free parameters for the cathode 1 current (G, C, µ, γ) and four for the grid current (Gg, Cg, ξ, Ig0). Surface Fitting Toolbox for MATLAB, The Mathworks

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6 →

2 anode current in mA 0 -5 -4 -3 -2 -1 0 1 2 3 20 60 100 140 180 220 260 300 grid voltage in V anode voltage in V → →

1.5 → 1

0.5

grid current in-0.5 mA -5 -4 -3 -2 -1 0 1 2 3 20 60 100 140 180 220 260 300 grid voltage in V anode voltage in V → → Figure 6: Complete characteristics of a 12AX7 triode. In black: simulation results, : measurement data from a RSD tube. Plots on the ◦ left: grid family for Va =100 V, 200 V and 300 V. Plots on the right: plate family for Vg = 2 V, 1 V, 0.1 V, 1 V and 2 V. − −

RSD-1 RSD-2 EHX-1 (black curves) and the measurements of one tube (circles). The G 2.242E-3 2.173E-3 1.371E-3 dotted curves show the characteristics of Koren’s model [10]. The µ 103.2 100.2 86.9 curves have generally the same progression. Differences are found γ 1.26 1.28 1.349 for the grid current (lower plots) and for the shape of the anode C 3.40 3.19 4.56 current for positive grid voltages. Gg 6.177E-4 5.911E-4 3.263E-4 This observation is not surprising. The new model was indi- ξ 1.314 1.358 1.156 vidually fitted to the measurement data, so the black curves should Cg 9.901 11.76 11.99 align better with the circles than the general Koren model. More Ig0 8.025E-8 4.527E-8 3.917 E-8 meaningful is the comparison to other individual models. We decided to examine the individual model proposed by Cohen and He- Table 1: Individually fitted parameters for 12AX7 triodes. lie [12]. It is based on Koren’s formula, but with a different grid current where a piecewise-defined function with three subdomains is suggested. To achieve an equitable comparison the equations 5.6. Parasitic Capacitances were fitted in a similar manner to the same measurement data. The results are displayed in Figure 7 as dashed curves. The improve- The electrodes and their mechanical assembly lead to parasitic ca- ment due to the individualization is clearly visible and the curves pacitances which have to be taken into account for a correct dy- align better with the measurements. namic behavior. As in other existing tube models we assume the Regarding our approach a deviation remains for high anode standard values that can be found in the data sheets, i.e. Cak = voltages and negative grid voltage (see upper left plot), where the 0.9 pF, Cgk = 2.3 pF and Cag = 2.4 pF for 12AX7. standard Koren model interestingly performs best. Besides, the shape for very low anode voltages (Va < 20 V) is not captured 6. RESULTS correctly. However, in this comparison the proposed equations allowed 6.1. Comparison with other Models better fitting results and achieved a good alignment with the mea- The proposed model has to be compared to the existing approaches. surements. The same observation was made for the other tested Figure 7 shows again the family characteristics of the new model triodes.

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6 →

2 anode current in mA 0 -5 -4 -3 -2 -1 0 1 2 3 20 60 100 140 180 220 260 300 grid voltage in V anode voltage in V → →

1.5 → 1

0.5

grid current in-0.5 mA -5 -4 -3 -2 -1 0 1 2 3 20 60 100 140 180 220 260 300 grid voltage in V anode voltage in V → →

Figure 7: Comparison of different triode models. Plots on the left: grid family for Va =100 V and 300 V. Plots on the right: plate family characteristics for Va = 2 V, 0.1 V and 2 V. Shown are Koren’s approach (dotted), Cohen and Helie (dashed) and the new model (black). The discrete measurement− points are marked with . ◦

Vb input amplitudes the grid will be temporarily positive. Several papers analyzed this simple but representative circuit and discussed Ra C out qualiﬁed simulation techniques [9, 12, 14, 15]. For the simulation of the circuit a state-space model was im- Rg plemented. The state-space representation has turned out to be V a a practical tool for the simulation of nonlinear circuits [16]. We skip the details of the implementation and refer to a recent study Vg Vin dealing with this method [17]. The state-space representation gets Rk Ck along with four state variables, two for the shown capacitors Cout and Ck and two for the parasitic capacitances of the triode. In fact, in Section 5.6 we introduced three capacitances (Cak, Cgk and Figure 8: Common-cathode ampliﬁer stage. Cag) but they are not independent and thus can be reduced to two.

6.2.1. Waveforms 6.2. Application: Common-Cathode Amplifier Stage A comparison of time signals is given in Figure 9, where the simi- To check to what extent the model equations replicate real triodes larity for different frequencies and input amplitudes is checked. As when used within a circuit simulation, a common tube amplifier expected, the waveforms for higher input amplitudes are distorted. stage was prototyped. The same tubes as those used in the fitting The measured and simulated outputs align well and only small dif- procedure were operated in this test circuit and the responses to ferences can be observed. All measurements and simulations were various excitation signals were measured. performed with a sampling frequency fs = 96 kHz. Figure 8 depicts the schematic of the chosen amplifier stage. This common-cathode amplifier can be found in almost all pream- 6.2.2. Harmonic Spectra plifiers designs. We skip a detailed circuit analysis and are content with the information, that this represents an inverting amplifier giv- Waveform comparisons give only limited information on how good ing a high gain. The operating point is a bit negative, but for higher a simulation performs. In addition, the harmonic content of the

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100 100 100 75 75 75 → → → 50 50 50 25 25 25 0 0 0 -25 -25 -25 amplitude in V amplitude in V amplitude in V -50 -50 -50 -75 -75 -75 -100 -100 -100

0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 time in ms time in ms time in ms → → → 100 100 100 75 75 75

→ 50 → 50 → 50 25 25 25 0 0 0 -25 -25 -25 amplitude in V amplitude in V amplitude in V -50 -50 -50 -75 -75 -75 -100 -100 -100

0 5 10 15 0 5 10 15 0 5 10 15 20 time in ms time in ms time in ms → → →

Figure 9: Waveform comparison of measurement (gray) and simulation (dashed black) for different input signals (solid black). First row: sinusoidal excitation with 500 Hz and 2 V, 4 V and 8 V. Second row: 4 V sine burst for frequencies 500 Hz, 1 kHz and 2 kHz. output is computed both for reference system and simulation. The nonlinear triode is the 12AT7, which can be found in audio circuits exponential sweep technique [18] was used to measure the intro- (e.g. phase inverter stages) as well. In addition to the 12AX7 mea- duced distortion. Figure 10(a) shows the harmonic distortions of surements, one 12AT7 triode was checked. It was found that the the discussed circuit for a 4 V measurement signal. The results of fitting results are not as good as for the 12AX7 tubes. Especially the simulation are given in 10(b). As the plots illustrate, a satisfy- the assumption that Ig is almost independent of Va is not valid ing similarity is achieved. Differences are visible at high frequen- anymore. As a second limitation we have to respect a lower bound cies, where the simulation is a bit flatter. for the anode voltage. When operating the triode model in the region with Vg > 0 and small anode voltages, say Va < 20 V, the results are obviously not correct. For real triodes the anode current 6.2.3. Guitar Sound will decrease rapidly when approaching Va = +0 V, an effect that Some sound clips of electric guitar playing are recorded with the is not reproduced by the presented formulations. circuit and simulated. The comparison supports the good confor- mance of the results. The clips are available on our homepage 7. DISCUSSION AND OUTLOOK http://ant.hsu-hh.de/dafx2011/tubemodel The proposed equations are able to characterize the behavior of 6.3. Limitations real triodes in a good accordance. One advantage over other de- scriptions is that the equations can be educed from the fundamental The measured type of triode 12AX7 is known as a linear tube [9]. laws to a large extent. Keeping guitar amplifier distortion in mind this may confuse at Though there are some restrictions as identified by the com- first. But the classification linear or nonlinear regime depends on parisons. As already stated, the shape for very low anode voltages the course of the amplification factor µ. A typical example for a (Va < 20 V) is not captured correctly. But normally the 12AX7

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0 0 → → -20 -20

|H| in-40 dB |H| in-40 dB

-60 -60

100 1k 10k 100 1k 10k frequency in Hz frequency in Hz → → (a) measurement (b) simulation.

Figure 10: Frequency domain analysis of the common-cathode amplifier. Smoothed harmonic spectra for a sweep with 4 V amplitude. Shown are the fundamental (black), odd (dashed gray) and even order harmonic responses (solid gray). triode will not be operated in this region, because the internal re- [7] R. Aiken, “What is blocking distortion?,” [online], sistance approaches a finite value for all anode voltages. Hence www.aikenamps.com/BlockingDistortion.html, 1999. this shortcoming is considered to be marginal. [8] M. Zollner, Physik der Elektrogitarre, (preprint), Regens- As a future prospect more tubes have to be inspected with even burg, 2010. higher resolution of the discrete measurement points to improve the quality of the fittings. Furthermore the model has to enhanced [9] A. Potchinkov, Simulation von Röhrenverstärkern mit so that nonlinear triodes are comprised, too. SPICE, Vieweg + Teubner, 1st edition, 2009. [10] N. Koren, “Improved vt models for spice simulations,” Glass 8. CONCLUSION Audio, vol. 5, pp. 18–27, 1996. [11] G. C. Cardarilli, M. Re, and L. Di Carlo, “Improved A new model for triodes of type 12AX7 was presented, featur- large-signal model for vacuum triodes,” in IEEE Interna- ing a good replication of the grid current and physically-motivated tional Symposium on Circuits and Systems (ISCAS), 2009, formulations. The equations are mainly based on the Langmuir- pp. 3006–3009. Child’s law and can be calculated separately for cathode and grid current. [12] I. Cohen and T. Helie, “Measures and parameter estimation Free parameters within the formulations were used to perform of triodes, for the real-time simulation of a multi-stage guitar an individual fitting to measurement data of practical triodes. It preamplifier,” in Proc. 129th AES Convention, San Fran- was shown, that the equations are able to characterize the proper- cisco, USA, Nov 4-7 2010, number 8219. ties of real tubes in good accordance. [13] H. Kniekamp, “Die Abweichungen der Verstärker- To prove the suitability for the simulation of audio circuits, röhrenkennlinien vom e3/2-Gesetz,” Telegraphen- und Fern- the model was embedded in a state-space description of a typical sprechtechnik, vol. 20, no. 3, pp. 71–76, 1931. tube preamplifier. The simulation results for different input ampli- [14] J. Macak and J. Schimmel, “Real-time guitar tube amplifier tudes and frequencies were compared to reference measurements simulation using an approximation of differential equations,” and showed a good match. The derived equations are simple, con- in Proc. of the 13th Int. Conference on Digital Audio Effects tinuously differentiable and applicable for real-time simulations. (DAFx-10), Graz, Austria, Sept. 6-10 2010. [15] F. Santagata, A. Sarti, and S. Tubaro, “Non-linear digital 9. REFERENCES implementation of a parametric analog tube ground cathode [1] H. Reich, Principles of Electron Tubes, McGraw Hill, 1st amplifier,” in Proc. of the 10th Int. Conference on Digital Au- edition, 1941. dio Effects (DAFx-07), Bordeaux, France, Sept. 10-15 2007. [2] H. Barkhausen, Lehrbuch der Elektronenröhren und ihrer [16] D.T. Yeh, Digital Implementation of Musical Distortion Cir- technischen Anwendungen, Verlag S. Hirzel, Leipzig, 1945. cuits by Analysis and Simulation, Ph.D. dissertation, Stan- ford University, June 2009. [3] K. Spangenberg, Vacuum Tubes, McGraw Hill, 1942. [17] K. Dempwolf, M. Holters, and U. Zölzer, “Discretization [4] H. Rothe and W. Kleen, Grundlagen und Kennlinien der of parametric analog circuits for real-time simulations,” in Elektronenröhren, Geest & Portig, Leipzig, 1951. Proc. of the 13th Int. Conference on Digital Audio Effects [5] J. Pakarinen and D. T. Yeh, “A review of digital techniques (DAFx-10), Graz, Austria, Sept. 6-10 2010. for modeling vacuum-tube guitar amplifiers,” Computer Mu- sic Journal, vol. 33, no. 2, pp. 85–100, Summer 2009. [18] A. Farina, “Simultaneous measurement of impulse response and distortion with a swept-sine technique,” in 108th AES [6] M. Blencowe, Designing Tube Preamps for Guitar and Bass, Convention, Paris, France, Feb. 19-24 2000. Blencowe, 2009.

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